Question #43146

A diffractive lens is placed at the origin ($z=0$) of an optical axis. It exhibits both longitudinal chromatic aberration and spherical aberration. The focal length $f$ for light of wavelength $\lambda$ incident at a radial distance $r$ from the optical axis is given by: $f(\lambda, r) = \frac{A}{\lambda} - k r^2$ where $A = 310 \text{ nm}\cdot\text{m}$ and $k = 50 \text{ m}^{-1}$.

A parallel beam of light containing a continuous spectrum of wavelengths from $50 \text{ nm}$ to $400 \text{ nm}$ is incident on the lens. An annular mask restricts the incident light to a thin ring of radius $r = 10 \text{ cm}$.

A thin wire photocathode is placed along the optical axis from $z = 1.0 \text{ m}$ to $z = 3.0 \text{ m}$. The work function of the wire varies with the distance $z$ from the lens as: $\phi(z) = (1.5 + 1.0 z^2) \text{ eV}$ where $z$ is in meters.

Assuming photoelectrons are only emitted from the exact focal spots on the wire, and taking $hc = 1240 \text{ eV}\cdot\text{nm}$, what is the maximum kinetic energy of the emitted photoelectrons?